q-Bernstein polynomials and Bézier curves

Journal of Approximation Theory, 2012

We introduce a new variant of the blossom, the q-blossom, by altering the diagonal property of the standard blossom. This q-blossom is specifically adapted to developing identities and algorithms for q-Bernstein bases and q-Bézier curves over arbitrary intervals. By applying the q-blossom, we generate several new identities including an explicit formula representing the monomials in terms of the q-Bernstein basis functions and a q-variant of Marsden's identity. We also derive for each q-Bézier curve of degree n, a collection of n! new, affine invariant, recursive evaluation algorithms. Using two of these new recursive evaluation algorithms, we construct a recursive subdivision algorithm for q-Bézier curves.

Journal of Approximation Theory, 2006

The evaluation of multivariate polynomials of n variables in Bernstein-Bézier form is considered. A forward error analysis for the corresponding de Casteljau algorithm and the VS algorithm is performed. We also include algorithms that simultaneously evaluate the polynomial and provide "a posteriori" error bounds, without increasing significantly the computational cost. The sharpness of our running error bounds is shown in the case of trivariate polynomials.

Journal of Approximation Theory

We introduce a new variant of the blossom, the q-blossom, by altering the diagonal property of the standard blossom. This q-blossom is specifically adapted to developing identities and algorithms for q-Bernstein bases and q-Bézier curves over arbitrary intervals. By applying the q-blossom, we generate several new identities including an explicit formula representing the monomials in terms of the q-Bernstein basis functions and a q-variant of Marsden's identity. We also derive for each q-Bézier curve of degree n, a collection of n! new, affine invariant, recursive evaluation algorithms. Using two of these new recursive evaluation algorithms, we construct a recursive subdivision algorithm for q-Bézier curves.

Journal of Computational and Applied Mathematics, 2002

The polynomials determined in the Bernstein (Bà ezier) basis enjoy considerable popularity and the algorithms for reducing their degree are of practical importance in computer aided design applications. On the other hand, the conversion between the Bernstein and the power basis is ill conditioned, thus only the degree reduction algorithms which may be carried out without using this conversion are of practical value. Our uniÿed approach enables us to describe all the algorithms of this kind known in the literature, to construct a number of new ones, which are better conditioned and cheaper than some of the currently used ones, and to study the errors of all of them in a simple homogeneous way. All these algorithms can be applied componentwise to a vector-valued polynomial representing a Bà ezier curve. Consider the values of the derivatives, whose orders vary successively from 1 to a given number à or Ä at the start and end point, respectively, of this curve. The current algorithms allow us to preserve these points and values for à equal to Ä, the new ones do that also without the latter constraint.

Journal of Computational and Applied Mathematics, 1994

In this paper a new algorithm for generating values of Bernstein-Bezier polynomials is proposed and investigated. Opposite to the known algorithms, its computational complexity does not depend on the degree of a calculated polynomial. However, it allows to calculate only approximate values of the polynomial. These features allow to recommend the proposed algorithm for solving curve fitting and image processing problems in which processing a large number of data is necessary. The proposed method is based on a probabilistic interpretation of Bernstein-Bezier (BB-) polynomials and its properties are investigated in the statistical language. As a byproduct, a local nature of BB-polynomial approximation is displayed.

2015

We present an efficient method to solve the problem of the constrained least squares approximation of the rational Bézier curve by the Bézier curve. The presented algorithm uses the dual constrained Bernstein basis polynomials, associated with the Jacobi scalar product, and exploits their recursive properties. Examples are given, showing the effectiveness of the algorithm.

For a given continuous function f (x) on [0, 1] we construct sequence of algebraic polynomials based on Bernstein approximation. We prove that the limit of this sequence is the Lagrange interpolation polynomial of degree n. Application to the representation of polynomial curves will be given.

Computer Aided Geometric Design, 2009

We present a novel approach to the problem of multi-degree reduction of Bézier curves with constraints, using the dual constrained Bernstein basis polynomials, associated with the Jacobi scalar product. We give properties of these polynomials, including the explicit orthogonal representations, and the degree elevation formula. We show that the coefficients of the latter formula can be expressed in terms of dual discrete Bernstein polynomials. This result plays a crucial role in the presented algorithm for multi-degree reduction of Bézier curves with constraints. If the input and output curves are of degree n and m, respectively, the complexity of the method is O (nm), which seems to be significantly less than complexity of most known algorithms. Examples are given, showing the effectiveness of the algorithm.

2017

In order to investigate the fundamental properties of q-Bernstein basis functions, we give generating functions for these basis functions and their functional and differential equations. In [16], [15] and [17], we construct a novel collection of generating functions to derive many known and some new identities for the classical Bernstein basis functions. The main purpose of this paper is to construct analogous generating functions for the q-Bernstein basis functions. By using an approach similar to that of our methods in [16] as well as some properties of interpolation functions, we can derive some known and some new identities, relations and formulas for the q-Bernstein basis functions, including the partition of unity property, formulas for representing the monomials, recurrence relations, formulas for derivatives, subdivision identities and integral representations. Furthermore, we give plots of not only our new basis functions, but also their generating functions. Also, we simul...

Results in Mathematics

We show that certain inequalities involving differences of the Bernstein basis polynomials and values of a function $$f\in C[0,1]$$ f ∈ C [ 0 , 1 ] , imply that the function is q-monotone. In view of previous results of the authors (see the list of references), the current results provide, among others, a characterization of q-monotone functions in C[0, 1].